For what value of x, is the matrix
`A=[(0,1,-2),(-1,x,-3),(2,3,0)]` a skew-symmetric matrix

Answer

To determine the value of \( x \) for which the matrix

\[ A = \begin{pmatrix} 0 & 1 & -2 \\ -1 & x & -3 \\ 2 & 3 & 0 \end{pmatrix} \]

is a skew-symmetric matrix, we follow these steps:

### Step 1: Understand the Definition of a Skew-Symmetric Matrix
A matrix \( A \) is skew-symmetric if \( A^T = -A \), where \( A^T \) is the transpose of \( A \).

### Step 2: Calculate the Transpose of Matrix A
The transpose of matrix \( A \) is obtained by swapping the rows and columns:

\[ A^T = \begin{pmatrix} 0 & -1 & 2 \\ 1 & x & 3 \\ -2 & -3 & 0 \end{pmatrix} \]

### Step 3: Set Up the Equation for Skew-Symmetry
According to the definition, we need:

\[ A^T = -A \]

Calculating \(-A\):

\[ -A = \begin{pmatrix} 0 & -1 & 2 \\ 1 & -x & 3 \\ -2 & -3 & 0 \end{pmatrix} \]

### Step 4: Equate the Elements of \( A^T \) and \(-A \)
Now, we equate the corresponding elements of \( A^T \) and \(-A \):

1. From the (1,2) position: \( -1 = -1 \) (True)
2. From the (1,3) position: \( 2 = 2 \) (True)
3. From the (2,1) position: \( 1 = 1 \) (True)
4. From the (2,2) position: \( x = -x \)
5. From the (2,3) position: \( 3 = 3 \) (True)
6. From the (3,1) position: \( -2 = -2 \) (True)
7. From the (3,2) position: \( -3 = -3 \) (True)

### Step 5: Solve the Equation \( x = -x \)
From the equation \( x = -x \), we can add \( x \) to both sides:

\[ x + x = 0 \implies 2x = 0 \implies x = 0 \]

### Conclusion
Thus, the value of \( x \) for which the matrix \( A \) is skew-symmetric is

\[ \boxed{0} \]

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